Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #8269 |
| commutator | C4×C8 | C2×C4×C8 |
| Frattini, cosocle | C4×#346 | C23 |
| socle | C2 | #72225528 |
| prime radical | C23 | #12667 |
| LCS | G ▷ C4×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#346 ◁ C4×#1398 ◁ G | |
| LECS | G ▷ C4×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,10,6,12,23,38,8,14,7,30,24,13,29,39)(3,9,16,31,19,33,21,34,20,36,46,59,18,35,15,44)(5,25,26,56,27,43,57,37,28,42,22,55,53,11,40,41)(17,48,49,62,51,32,45,60,52,61,47,64,50,63,54,58);(1,3,5,17,7,15,26,45,6,18,27,50,29,46,57,49,8,20,28,52,4,21,22,54,24,19,53,51,23,16,40,47)(2,9,11,32,10,34,37,62,12,35,25,48,38,31,55,58,14,36,43,63,30,44,41,64,13,33,42,61,39,59,56,60);(2,10,12,38,14,30,13,39)(3,15,18,46,20,21,19,16)(11,37,25,55,43,41,42,56)(17,45,50,49,52,54,51,47)Polycyclic presentation:
2^11|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=ijk,g^2=k,h^2=j,i^2=k,j^2=k,b^a=bf,c^a=cfhk,c^b=cfhijk,d^b=djk,e^a=ej,e^d=ek,f^a=fi,f^b=fi,g^b=gk,h^a=hk,i^a=ik,i^b=ik176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225554, #72227066, #72227077, … |
| maximal normal subgroup of | #72225554, #72227066, #72227077, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.10*F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);